Group #34206877 — subgroups

Subgroups — #34206877

Order 20224.

374 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 159 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098965	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098339	C₂	1	1	yes	maximal, characteristic
3	#23761491	C₂	1	4	yes	maximal
4	C₇₉×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579153	—	2	1	no	—
6	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
7	#19098965	—	2	2	no	—
8	#19098965	C₄	1	4	yes	complemented
9	C₂×#14579095	C₂²	1	2	yes	—
10	C₂×#14579093	—	2	1	no	—
11	#19098339	—	2	2	no	—
12	C₂×#107×C₇₉	—	2	1	no	—
13	C₂×#107×C₇₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
15	#384×C₇₉	—	2	1	no	—
16	#14579153	—	2	4	no	—
17	#14579153	—	4	2	no	—
18	C₂×C₄⋊Dic₇₉	—	2	2	no	—
19	C₂×Dic₇₉⋊C₄	—	4	2	no	—
20	C₂×C₇₉⋊C₁₆	—	2	2	no	—
21	C₂×C₇₉⋊C₁₆	—	2	2	no	—
22	C₂×C₇₉⋊C₁₆	Q₈	1	4	yes	—
23	#14579095	—	2	2	no	—
24	#14579095	C₂×C₄	1	4	yes	—
25	#14579093	—	2	4	no	—
26	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₇₉	—	4	2	no	—
30	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
31	#107×C₇₉	—	2	4	no	—
32	#107×C₇₉	—	4	2	no	—
33	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₇₉	D₄	1	2	yes	—
35	C₂²×Dic₇₉	—	4	2	no	—
36	C₄⋊Dic₇₉	—	2	4	no	—
37	C₄⋊Dic₇₉	—	4	2	no	—
38	Dic₇₉⋊C₄	—	4	8	no	—
39	C₇₉⋊C₁₆	—	2	4	no	—
40	C₂²×C₄×C₇₉	—	4	2	no	—
41	C₂²×C₄×C₇₉	D₈	1	2	yes	—
42	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
44	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
45	C₂×C₈×C₇₉	Q₁₆	1	2	yes	—
46	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
49	C₂×C₈×C₇₉	#41	1	2	yes	—
50	C₄⋊C₄×C₇₉	—	4	2	no	—
51	C₄⋊C₄×C₇₉	—	4	2	no	—
52	C₄⋊C₄×C₇₉	—	8	4	no	—
53	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
54	C₄²×C₇₉	SD₁₆	1	2	yes	—
55	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₇₉	—	2	2	no	—
57	C₂×C₄×C₇₉	—	2	2	no	—
58	C₂×C₄×C₇₉	—	8	2	no	—
59	C₂×C₄×C₇₉	—	8	4	no	—
60	C₂×C₄×C₇₉	#102	1	4	yes	—
61	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
63	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×Dic₇₉	—	4	4	no	—
68	C₂×Dic₇₉	—	8	2	no	—
69	C₂×Dic₇₉	—	8	2	no	—
70	C₈×C₇₉	#103	1	4	yes	—
71	C₈×C₇₉	C₄×Q₈	1	2	yes	—
72	C₈×C₇₉	C₂²⋊Q₈	1	2	yes	—
73	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
74	C₂²×C₇₉	#486	1	1	yes	characteristic
75	C₂²×C₇₉	#500	1	1	yes	characteristic
76	C₂²×C₇₉	#509	1	2	yes	—
77	C₂²×C₇₉	#510	1	2	yes	—
78	C₄×C₇₉	—	2	4	no	—
79	C₄×C₇₉	—	16	2	no	—
80	C₄×C₇₉	#391	1	1	yes	characteristic
81	C₄×C₇₉	C₂×#103	1	1	yes	characteristic
82	C₄×C₇₉	#487	1	1	yes	characteristic
83	C₄×C₇₉	#498	1	1	yes	characteristic
84	Dic₇₉	—	8	4	no	—
85	#13284	—	79	1	no	maximal, Sylow
86	C₂×C₇₉	#1990	1	1	yes	characteristic
87	C₂×C₇₉	#2139	1	2	yes	—
88	C₂×C₇₉	#2244	1	1	yes	characteristic
89	C₂×C₇₉	#2245	1	1	yes	characteristic
90	C₂×C₇₉	#2255	1	2	yes	—
91	C₂×#499	—	79	1	no	—
92	C₂×#346	—	79	1	no	—
93	#1845	D₇₉	1	1	yes	characteristic
94	#1380	—	79	4	no	—
95	C₇₉	#13284	1	1	yes	characteristic, complemented, Sylow
96	C₂×#125	—	158	1	no	—
97	C₂²×C₁₆	—	79	2	no	—
98	#499	—	79	4	no	—
99	#499	—	158	2	no	—
100	C₂×#107	—	2	1	no	—
101	C₂×#107	—	79	2	no	—
102	C₂×#107	C₂×D₇₉	1	2	yes	—
103	C₂×#106	—	158	1	no	—
104	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
105	#384	—	2	1	no	—
106	#346	—	158	2	no	—
107	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
108	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
109	#125	—	316	2	no	—
110	#125	—	158	4	no	—
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄⋊C₄	—	4	2	no	—
113	C₂×C₄⋊C₄	—	316	2	no	—
114	C₂×C₄⋊C₄	—	158	2	no	—
115	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
116	C₂×C₁₆	—	79	4	no	—
117	C₂×C₁₆	—	158	2	no	—
118	C₂×C₁₆	—	158	2	no	—
119	#107	—	2	4	no	—
120	#107	—	4	2	no	—
121	#107	—	79	4	no	—
122	#107	—	158	2	no	—
123	#106	—	158	4	no	—
124	C₄×C₈	C₄×D₇₉	1	2	yes	—
125	C₄×C₈	D₃₁₆	1	2	yes	—
126	C₂²×C₄	—	4	2	no	—
127	C₂²×C₄	—	316	2	no	—
128	C₂²×C₄	#11929609	1	2	yes	—
129	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
130	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
131	C₂×C₈	#11929612	1	2	yes	—
132	C₂×C₈	#11929614	1	2	yes	—
133	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
134	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
135	C₂×C₈	Q₈×D₇₉	1	2	yes	—
136	C₂×C₈	#11929623	1	2	yes	—
137	C₄⋊C₄	—	4	2	no	—
138	C₄⋊C₄	—	4	2	no	—
139	C₄⋊C₄	—	8	4	no	—
140	C₄⋊C₄	—	316	8	no	—
141	C₄⋊C₄	—	316	2	no	—
142	C₄⋊C₄	—	158	4	no	—
143	C₄²	(C₂×D₇₉)⋊C₄	1	2	yes	—
144	C₄²	#11929611	1	2	yes	—
145	C₁₆	—	158	4	no	—
146	C₂³	C₂×#11929609	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	8	2	no	—
150	C₂×C₄	—	8	4	no	—
151	C₂×C₄	—	316	4	no	—
152	C₂×C₄	—	632	2	no	—
153	C₂×C₄	—	632	2	no	—
154	C₂×C₄	#14579081	1	4	yes	—
155	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
156	C₂×C₄	#14579148	1	1	yes	characteristic
157	C₂×C₄	#14579159	1	1	yes	characteristic
158	C₂×C₄	C₂×#11929611	1	1	yes	characteristic
159	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
160	C₂×C₄	C₂×#11929614	1	1	yes	characteristic
161	C₈	#14579084	1	4	yes	—
162	C₈	#14579126	1	2	yes	—
163	C₈	#14579141	1	2	yes	—
164	C₂²	C₂×#14579081	1	1	yes	characteristic
165	C₂²	#19098909	1	1	yes	characteristic
166	C₂²	#19098966	1	1	yes	characteristic
167	C₂²	#19098999	1	2	yes	—
168	C₂²	#19099004	1	2	yes	—
169	C₄	—	2	4	no	—
170	C₄	—	16	2	no	—
171	C₄	—	632	4	no	—
172	C₄	#19098468	1	1	yes	characteristic
173	C₄	C₂×#14579084	1	1	yes	characteristic
174	C₄	#19098913	1	1	yes	characteristic
175	C₄	#19098959	1	1	yes	characteristic
176	C₂	#23764360	1	1	yes	characteristic
177	C₂	#23765110	1	2	yes	—
178	C₂	#23765641	1	1	yes	characteristic
179	C₂	#23765645	1	1	yes	characteristic
180	C₂	#23765683	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19098965

C₂

yes

maximal, characteristic

C₂×#19098339

C₂

yes

maximal, characteristic

#23761491

C₂

yes

maximal

C₇₉×#1845

C₂

yes

maximal, characteristic

C₂×#14579153

—

C₂²×C₇₉⋊C₁₆

C₂²

yes

—

#19098965

—

#19098965

C₄

yes

complemented

C₂×#14579095

C₂²

yes

—

C₂×#14579093

—

#19098339

—

C₂×#107×C₇₉

—

C₂×#107×C₇₉

C₂²

yes

—

C₂×C₄×C₈×C₇₉

C₂²

yes

characteristic

#384×C₇₉

—

#14579153

—

#14579153

—

C₂×C₄⋊Dic₇₉

—

C₂×Dic₇₉⋊C₄

—

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

Q₈

yes

—

#14579095

—

#14579095

C₂×C₄

yes

—

#14579093

—

C₂²×C₈×C₇₉

D₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄⋊C₄×C₇₉

—

C₂×C₄²×C₇₉

D₄

yes

characteristic

#107×C₇₉

—

#107×C₇₉

—

C₄×C₈×C₇₉

C₂×C₄

yes

—

C₄×C₈×C₇₉

D₄

yes

—

C₂²×Dic₇₉

—

C₄⋊Dic₇₉

—

C₄⋊Dic₇₉

—

Dic₇₉⋊C₄

—

C₇₉⋊C₁₆

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

D₈

yes

—

C₂²×C₄×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

SD₁₆

yes

—

C₂×C₈×C₇₉

Q₁₆

yes

—

C₂×C₈×C₇₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×Q₈

yes

—

C₂×C₈×C₇₉

#41

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄²×C₇₉

C₂²⋊C₄

yes

—

C₄²×C₇₉

SD₁₆

yes

—

C₂³×C₇₉

C₂×D₈

yes

characteristic

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

#102

yes

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

#124

yes

characteristic

C₂×C₄×C₇₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₉

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₇₉

C₂×Q₁₆

yes

characteristic

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₂×Dic₇₉

—

C₈×C₇₉

#103

yes

—

C₈×C₇₉

C₄×Q₈

yes

—

C₈×C₇₉

C₂²⋊Q₈

yes

—

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

C₂²×C₇₉

yes

—

C₂²×C₇₉

#510

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

Dic₇₉

—

—

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×C₇₉

yes

C₂×C₇₉

yes

C₂×C₇₉

yes

—

C₂×#499

—

C₂×#346

—

#1845

D₇₉

yes

characteristic

#1380

—

C₇₉

#13284

yes

characteristic, complemented, Sylow

C₂×#125

—

158

—

C₂²×C₁₆

—

#499

—

#499

—

158

—

100

C₂×#107

—

101

C₂×#107

—

102

C₂×#107

C₂×D₇₉

yes

—

103

C₂×#106

—

158

—

104

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

105

#384

—

106

#346

—

158

—

107

C₂²×C₈

C₂²⋊D₇₉

yes

characteristic

108

C₂²×C₈

C₂²×D₇₉

yes

characteristic

109

#125

—

316

—

110

#125

—

158

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

316

—

114

C₂×C₄⋊C₄

—

158

—

115

C₂×C₄²

C₂²⋊D₇₉

yes

characteristic

116

C₂×C₁₆

—

117

C₂×C₁₆

—

158

—

118

C₂×C₁₆

—

158

—

119

#107

—

120

#107

—

121

#107

—

122

#107

—

158

—

123

#106

—

158

—

124

C₄×C₈

C₄×D₇₉

yes

—

125

C₄×C₈

D₃₁₆

yes

—

126

C₂²×C₄

—

127

C₂²×C₄

—

316

—

128

C₂²×C₄

#11929609

yes

—

129

C₂²×C₄

C₂×C₂²⋊D₇₉

yes

characteristic

130

C₂×C₈

(C₂×D₇₉)⋊C₄

yes

—

131

C₂×C₈

#11929612

yes

—

132

C₂×C₈

#11929614

yes

—

133

C₂×C₈

C₂×C₄×D₇₉

yes

characteristic

134

C₂×C₈

C₂×D₃₁₆

yes

characteristic

135

C₂×C₈

Q₈×D₇₉

yes

—

136

C₂×C₈

#11929623

yes

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

316

—

141

C₄⋊C₄

—

316

—

142

C₄⋊C₄

—

158

—

143

C₄²

(C₂×D₇₉)⋊C₄

yes

—

144

C₄²

#11929611

yes

—

145

C₁₆

—

158

—

146

C₂³

C₂×#11929609

yes

characteristic

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

316

—

152

C₂×C₄

—

632

—

153

C₂×C₄

—

632

—

154

C₂×C₄

#14579081

yes

—

155

C₂×C₄

C₂×(C₂×D₇₉)⋊C₄

yes

characteristic

156

C₂×C₄

#14579148

yes

characteristic

157

C₂×C₄

#14579159

yes

characteristic

158

C₂×C₄

C₂×#11929611

yes

characteristic

159

C₂×C₄

C₂×#11929612

yes

characteristic

160

C₂×C₄

C₂×#11929614

yes

characteristic

161

C₈

#14579084

yes

—

162

C₈

#14579126

yes

—

163

C₈

#14579141

yes

—

164

C₂²

C₂×#14579081

yes

characteristic

165

C₂²

#19098909

yes

characteristic

166

C₂²

#19098966

yes

characteristic

167

C₂²

#19098999

yes

—

168

C₂²

#19099004

yes

—

169

C₄

—

170

C₄

—

171

C₄

—

632

—

172

C₄

#19098468

yes

characteristic

173

C₄

C₂×#14579084

yes

characteristic

174

C₄

#19098913

yes

characteristic

175

C₄

#19098959

yes

characteristic

176

C₂

#23764360

yes

characteristic

177

C₂

#23765110

yes

—

178

C₂

#23765641

yes

characteristic

179

C₂

#23765645

yes

characteristic

180

C₂

#23765683

yes

—

	subgroup	G/H
center	C₂³	C₂×#11929609
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1845	C₂
socle, prime radical	C₂³×C₇₉	C₂×D₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#1845	D₇₉
hypocenter	C₇₉	#13284
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

subgroup

G/H

center

C₂³

C₂×#11929609

commutator

C₂×C₈×C₇₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#1845

C₂

socle, prime radical

C₂³×C₇₉

C₂×D₈

cosocle

C₂²×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

N	H ≅ G/N	type
8. #19098965	C₄	semidirect
95. C₇₉	#13284	semidirect

H ≅ G/N

type

8. #19098965

C₄

semidirect

95. C₇₉

#13284

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#19098965	2
C₂×#19098339	2
#23761491	2
C₇₉×#1845	2
#13284	79

subgroup

index

C₇₉×#1845

Subgroups — #34206877

Conjugacy classes of subgroups

Complements

Maximal subgroups